Vygantas Paulauskas

The Accuracy of Gaussian Approximation in Banach Spaces

Let B be a real separable Banach space with norm || · || = || · || B . Suppose that X, X 1, X 2, … ∈ B are independent and identically distributed (i.i.d.) random elements (r.e.’s) taking values in B. Furthermore, assume that EX = 0 and that there exists a zero-mean Gaussian r.e. Y ∈ B such that the covariances of X and Y coincide.

On Beveridge-Nelson decomposition and limit theorems for linear random fields

We consider linear random fields and show how an analogue of the Beveridge-Nelson decomposition can be applied to prove limit theorems for sums of such fields.

A New Estimator for a Tail Index

We investigate properties of a new estimator for a tail index introduced by Davydov and co-workers. The main advantage of this estimator is the simplicity of the statistic used for the estimator. We provide results of simulation by comparing plots of ours and Hills estimators.

On the Estimation of the Parameters of Multivariate Stable Distributions

In the paper, the asymptotic normality for a new estimator for the spectral measure of a multivariate stable distribution is proved. Also an estimator for the density of a multivariate stable distribution is proposed, its properties are investigated. The dependence of a stable density on exponent α and the spectral measure is investigated.

On operator-norm approximation of some semigroups by quasi-sectorial operators

Abstract We use some results and methods of the probability theory to improve bounds for the convergence rates in some approximation formulas for operators.

Some Asymptotic Results for One-Sided Large Deviation Probabilities

We investigate the asymptotic behavior of the sum of independent real random variables. We assume that the random variables are not identically distributed but the average of distribution functions of these random variables is equivalent to some heavy-tailed limit distribution function. An example with Pareto law as limit function is given.

On the central limit theorem in D[0, 1]

Let X be a stochastically continuous random process with sample paths in D[0, 1]. We improve Billingsleys theorem and this improvement yields new sufficient conditions for X to satisfy the CLT. An example shows that our sufficient conditions are close to the optimal conditions of the form provided.

Bounds for the accuracy of Poissonian approximations of stable laws

Stable law Gz admit a well-known series representation of the type where [Gamma]1, [Gamma]2, ... are the successive times of jumps of a standard Poisson process, and X1, X2, ..., denote i.i.d. random variables, independent of [Gamma]1, [Gamma]2, ... We investigate the rate of approximation of G[alpha] by distributions of partial sums Sn = [summation operator]nj = 1 [Gamma]-1/[alpha]jXj, and we get (asymptotically) optimal bounds for the variation of . The results obtained complement and improve the results of A. Janicki and P. Kokoszka, and M. Ledoux and V. Paulauskas. Bounds for the concentration function of Sn are also proved.

Statistical inference in regression with heavy-tailed integrated variables

We consider the problem of statistical inference in a bivariate time series regression model when the innovations are heavy-tailed and the OLS estimator is used for parameter estimation. We develop the asymptotic theory for the OLS estimator and the corresponding t-statistics. Limit distributions, that enable us to construct confidence intervals for the estimated parameters, are obtained via Monte Carlo simulations. The approach allows the components of the innovation vector to have different tail behavior.

Lévy–LePage Series Representation of Stable Vectors: Convergence in Variation

AbstractMultidimensional stable laws Gα admit a well-known Lévy–LePage series representation